Initial analysis and optimization of Edwards-curve operation counts
by Daniel J. Bernstein and Tanja Lange
suggested that Edwards curves would save time
in many elliptic-curve computations,
including ECM.
See https://hyperelliptic.org/tanja/newelliptic/.

GMP-EECM, a variant of GMP-ECM,
was the first software to set speed records using Edwards curves.
EECM-MPFQ, the current EECM software, is even faster.
EECM, GMP-EECM, and EECM-MPFQ were introduced in the following paper:

Acknowledgments

Daniel J. Bernstein
(https://cr.yp.to/djb.html)
was supported by
the U.S. National Science Foundation, grant number ITR-0716498.
"Any opinions, findings, and conclusions or recommendations
expressed in this material
are those of the author(s)
and do not necessarily reflect the views of the National Science Foundation."